*2.4. Classical Equations*

The flow formulae of viscous and steady Prandtl–Eyring nanofluid (P-ENF) in combination with variant thermal conductivity, radiation, and porous material are [31–33].

$$\frac{\partial G\_1}{\partial x} + \frac{\partial G\_2}{\partial y} = 0,\tag{3}$$

$$\mathbf{G}\_{1}\frac{\partial \mathbf{G}\_{1}}{\partial \mathbf{x}} + \mathbf{G}\_{2}\frac{\partial \mathbf{G}\_{1}}{\partial \mathbf{y}} = \frac{A\_{p}}{\mathbb{C}\rho\_{nf}} \left(\frac{\partial^{2} \mathbf{G}\_{1}}{\partial \mathbf{y}^{2}}\right) - \frac{A\_{p}}{2\mathbf{C}^{3}\rho\_{nf}}\frac{\partial^{2} \mathbf{G}\_{1}}{\partial \mathbf{y}^{2}} \left[\left(\frac{\partial \mathbf{G}\_{1}}{\partial \mathbf{y}}\right)^{2}\right] - \frac{\mu\_{nf}}{\rho\_{nf}k}\mathbf{G}\_{1\prime} \tag{4}$$

$$G\_1 \frac{\partial \Psi}{\partial x} + G\_2 \frac{\partial \Psi}{\partial y} = \frac{1}{\left(\rho \mathbb{C}\_p\right) \kappa\_{nf}} \left[\frac{\partial}{\partial y} \left(\kappa\_{mf}^\*(\mathbb{A}^\circ) \frac{\partial \mathbb{A}^\circ}{\partial y}\right)\right] - \frac{1}{\left(\rho \mathbb{C}\_p\right)\_{nf}} \left[\frac{\partial q\_r}{\partial y}\right],\tag{5}$$

the appropriate connection conditions were as follows (Aziz et al. [34]):

$$\mathbf{G}\_{1}(\mathbf{x},0) = l\mathbf{I}\_{\overline{w}} + \mathbf{N}\_{L} \left(\frac{\partial \mathbf{G}\_{1}}{\partial y}\right), \quad \mathbf{G}\_{2}(\mathbf{x},0) = V\_{\overline{\pi}\prime} \quad -k\_{\overline{\pi}} \left(\frac{\partial \overline{\mathbf{x}}}{\partial y}\right) = h\_{\overline{\pi}}(\mathbf{\overline{\pi}\_{w}} - \mathbf{\overline{x}}), \tag{6}$$

$$G\_1 \to 0, \quad \forall \to \: \mathbb{F}\_{\infty} \text{ as } y \to \infty. \tag{7}$$


Other crucial parameters involved fluid parameters *Ap*, C, slip length *NL*, surface permeability *V<sup>π</sup>*, heat transfer coefficient *h<sup>π</sup>*, and porosity (*k*), along with heat conductivity of firm *k<sup>π</sup>*. It considered physical elements such as the thermal loss from a conventionally heated surface due to conduction and velocity at the surface as a function of the shear stress applied to it (slip condition). Because of the thickness of non-Newtonian P-ENF, just a short distance was covered by the radiative flow. Therefore, radiation heat flux estimation obtained through Rosseland [35] was applied in Equation (5).

$$\eta\_r = -\frac{4\sigma^\*}{3k^\*} \frac{\partial \Psi^4}{\partial y}\_{,} \tag{8}$$

herein, *σ*<sup>∗</sup> represents the Stefan–Boltzmann constant. Table 1 summarizes the equations of P-ENF material variables [36,37]:

**Table 1.** Formulae used for studied nanofluids [36,37].


*φ* represents the volume fraction coefficient of nanofluid. *μf* , *ρf* , *κ f* and (*Cp*)*f* show dynamic viscosity, density, thermal conductivity, and functional heat capacity regarding the ideal fluid, respectively. The indice of "s" represents the solid nanoparticles. (*κ*<sup>∗</sup>*n f*(-)) represents the temperature-reliant heat conductance of nanofluid.

The thermophysical properties of engine oil and studied nanoparticles are shown in Table 2 [38,39].

**Table 2.** Materials thermophysical properties [38,39].

